Section 10.15

Power Series and Functions

Turning complex functions into infinite polynomials.

Introduction

Why stop at ? We can represent many functions, like or , as power series. This allows us to integrate "impossible" functions and solve differential equations easily.

Interactive: Approximating Logarithms

As increases, the series hugs tighter, but only between -1 and 1.

The Geometric Series Trick

Start with this:

You can replace "" with anything!
Example: Replace with :
.

Differentiation & Integration

Since , we can integrate the series for to get .

Key Fact:

Term-by-term differentiation and integration preserves the Radius of Convergence (though endpoints might change).

Worked Examples

Example 1: Rational Function

Represent as a series.

Geometric form: .
Substitute for :
.

Valid for .

Example 2: Arctan(x)

Find series for .

Start with derivative: .
Integrate term-by-term:
.

Solve C using .
Final: .